3. Vilhelm, who has been physically inactive his whole life, decides to complete a triathlon with the aim that his finis

[answerhappygod]

3. Vilhelm, who has been physically inactive his whole life,
decides to complete a triathlon with the aim that his finishing
time puts him in the top half of the field. Vilhelm considers two
different strategies for training: the soft option and the hard
option. Compared with using the soft option using the hard option
will improve Vilhelm’s chances of finishing in the top half, but it
will also increase his probability of getting seriously injured
while training. If Vilhelm gets seriously injured he won’t be able
to complete the race at all.
• Vilhelm estimates that using the hard option leads to a 10%
chance that he gets seriously injured, a 60% chance of finishing in
the top half, and a 30% chance that he finishes in the bottom
half.
• On the other hand, he estimates that using the soft option leads
to a 1% risk of serious injury, a 30% chance of finishing in the
top half and a 69% chance of finishing in the bottom half.
After having thought long and hard about it Vilhelm still can’t
pick a training strategy. So he decides to roll a fair six-sided
die to decide, if the die shows a number less than or equal to 4 he
will use the soft option, otherwise he will use the hard
option.
• Let S be the event that Vilhelm uses the soft way, and H the
event that he uses the hard way.
• Let C be the event that Vilhelm completes the triathlon.
• Let I be the event that Vilhelm gets seriously injured.
• Let W be the event that Vilhelm’s time is among the top half of
participants.
(a)
Basedontheaboveinformation,writedownP(I|H),P(W|H),P(Wc|H),P(I|S),P(W|S),
and P(Wc |S).
(b) Are events I and W mutually exclusive events? Justify your
answer.
(c) Find P(W ∩S) and P(W ∩H).
(d) What is the chance that Vilhem finishes in the top half?
(e) Can we represent the probability that “Vilhelm used the hard
method given that he com-
pleted the race” with P(H|Ic)? Justify your answer.
(f) FindP(C∩W).
(g) Find P(C|H) and P(C|S).
(h) Find the probability that Vilhelm completes the triathlon, i.e.
find P(C).
(i) What is the probability that Vilhelm used the hard way given
that he completed the race and his time was among the top half of
finishing times is: